Multifocal optical devices find utility in a number of technologies. For example, bifocal and trifocal lenses treat presbyopia caused by loss of accommodative power of the eye. Progressive addition lenses for this purpose include a far-view zone with a low power for distance vision and a near-view zone with higher power for reading and other close work. The power increases progressively and smoothly in an intermediate zone or corridor between the far and near zones. Normally, the near zone lies near the bottom center of the lens, below the far zone; however, other arrangements satisfy specialized purposes.
Lens design generates a specification of surface points that a grinder can produce from a lens blank. Modern lens grinders are numerically controlled machines that abrade the front and/or back surfaces of the blank according to design data specifying various points on the surface. Some lenses may alternatively be produced by plastic casting or other techniques. Both the front and back surfaces of a lens may have magnifying power; the effective power is approximately the difference between the powers of the two surfaces. Power at any point is related to the sum of a surface's curvatures in two perpendicular directions.
Astigmatism is normally an undesirable property of lenses that causes distortion and color fringing. Astigmatism, sometimes called “cylinder,” is related to the difference between the lens curvatures in different directions. While a single power lens can theoretically have zero astigmatism everywhere, a lens having different powers at different locations necessarily has at least some astigmatism. A major goal in designing progressive lenses is to reduce astigmatism, especially in the region of the far, near, and intermediate zones. Secondary zones outside this region are less critical, although astigmatism reduction there is still important.
Computer-aided design of progressive lens and similar devices have employed both direct and indirect methods. A direct method first prescribes a selected distribution of curvatures on a vertical meridian line that follows the eye's up-down motion. Prescribed horizontal curves across the meridian are chosen to have the desired curvatures where they cross the meridian. Direct methods usually result in a high astigmatism level with little control over the astigmatism in different areas of the lens surface.
Indirect methods balance a desired power distribution with the unavoidable astigmatism produced. A variational approach iterates toward the solution of a full non-linear fourth-order Euler-Lagrange partial differential equation. Another approach attempts to minimize a defined cost function with an optimization algorithm. The cost function attains a low value when the desired lens properties are approached. One such method specifies a desired power distribution over an entire lens surface, then constructs a surface close to this distribution while optimizing to achieve low weighted amounts of astigmatism. Free-energy analogies and variational approaches have also been investigated. Indirect methods involve the solution of fourth-order nonlinear elliptic equations, which involve very large amounts of computation—up to several hours for a single lens design.
A lens design may deliberately include an astigmatic or cylinder component to correct for astigmatism in the wearer's eyes. Integrating the magnifying-power correction and the astigmatism or cylinder correction in the same surface of a lens or other optical device is desirable. Single-power lens conventionally fabricate both corrections in the same surface of a toric (or atoric) lens. However, progressive design techniques have heretofore not integrated astigmatism correction, and fabricating the two corrections sequentially on the same surface would unacceptably thin the lens or device. Conventional methods therefore fabricate the progressive component on one surface (usually the front), and the astigmatism correction on the other surface. Although this makes the two corrections approximately independent of each other, some offset errors still exist, and structural weakness may still occur, especially in currently fashionable thin, light ophthalmic lenses.